A mathematical analysis of a model of structured population. I. (English) Zbl 1261.47062

The author considers a kinetic model of structured populations \[ \frac {\partial f}{\partial t}+\frac {\partial f}{\partial a}+\mu (a,l)f(t,a,l)=P_{\eta }f=\int _{0}^{l}\int _{0}^{a}\eta (a,l,a^{\prime },l^{\prime })f(t,a^{\prime },l^{\prime })\text{d}a^{\prime }\text{d}l^{\prime } \] with initial data in \(L^{1}(\Omega ),\) where \( \Omega =\left \{ (a,l): 0<a<l, \;0<l<l_{2}\right \} \) with “boundary” conditions \(f(t,0,.)=K\tilde {f}(t)\), where \(\tilde {f}(t)\:l\in (0,l_2)\rightarrow f(t,l,l)\) and the boundary operator \(K\) is linear and bounded on \(L^1(0,l_{2})\) with norm \( \left \| K\right \| \geqslant 1\). The main concern of the paper is the well-posedness of this equation in \(L^{1}\) spaces in the sense of semigroup theory. Since \(\mu \in L^{\infty }(\Omega )\) and \(P_{\eta }\in \mathcal {L} (L^{1}(\Omega ))\), without loss of generality, the author restricts himself to the case \(\mu =0\) and \(P_{\eta }=0\). (We point out that generation theory is standard when \(\left \| K\right \| <1\) for general divergence free vector fields and the resulting semigroups are contractive; see, e.g., [W. Greenberg et al., Boundary value problems in abstract kinetic theory. Basel-Boston-Stuttgart: Birkhäuser Verlag (1987; Zbl 0624.35003)], Chapter 12.)
The author shows that, if \(1_{\varepsilon }\) denotes the multiplication operator by the indicator function of the interval \(\left ( 0,\varepsilon \right ) \) and if \(\lim _{\varepsilon \rightarrow 0}\left \| K1_{\varepsilon }\right \| _{\mathcal {L}(L^{1}(\Omega ))}<1\), then the unbounded operator \(- \frac {\partial f}{\partial a}\) on \(L^{1}(\Omega )\) with boundary condition \( f(0,.)=K\tilde {f}\) (where \(\tilde {f}\) is the trace of \(f\) on \( \left \{ (l,l): 0<l<l_{2}\right \}\)) generates a positive semigroup. The proof consists in using a change of unknown which transforms (in an equivalent way) the initial problem into another one with boundary operator with norm \(<1\). This result is already known in all \(L^{p}(\Omega )\) spaces (\(1\leq p<+\infty \)) with essentially the same proof (see Theorem 3.2 and Theorem 3.3 in [B. Lods and M. Mokhtar-Kharroubi, J. Math. Anal. Appl. 266, No. 1, 70–99 (2002; Zbl 1056.92047)]). Moreover, this result holds for general divergence free vector fields in very general geometries (see Theorem 5.1 in [L. Arlotti et al., Mediterr. J. Math. 8, No. 1, 1–35 (2011; Zbl 1230.47073)]).
To provide concrete generation theorems, the author attempts to show (in his Proposition 3.1) that, if \(K=K_{b}+K_{c}\), where \(\left \| K_{b}\right \| <1\) and \(K_{c}\) is a compact operator, then \[ \lim _{\varepsilon \rightarrow 0}\left \| K1_{\varepsilon }\right \| _{\mathcal {L}(L^{1}(\Omega ))}<1. \] The generation result behind this is probably true, this is the case in \(L^{p}(\Omega )\) spaces with \(1<p<+\infty \) (see Corollary 3.1 in [B. Lods and M. Mokhtar-Kharroubi, J. Math. Anal. Appl. 266, No. 1, 70–99 (2002; Zbl 1056.92047)]), but the proof of the author is, alas, erroneous. Indeed, using the unit ball \(B(0,1)\) of \(L^{1}(0,l_{2})\) and the precompactness of \( K_{c}1_{\varepsilon }(B(0,1))\), for any \(\alpha >0\) the author recovers that \( K_{c}1_{\varepsilon }(B(0,1))\) can be covered by a finite number \(n_{\alpha }\) of balls \(B(K_{c}1_{\varepsilon }(\psi _{l}),\alpha )\), \(1\leq l\leq n_{\alpha } \) with radius \(\alpha \) centered at \(K_{c}1_{\varepsilon }(\psi _{l})\) with \(\psi _{l}\in B(0,1)\) and (forgetting that \(n_{\alpha }\) and the \(\psi _{l}^{\prime }s\) should depend also on \(\varepsilon\)) passes to the limit as \(\varepsilon \rightarrow 0\) to arrive at \(\left \| K_{c}1_{\varepsilon }\right \| \rightarrow 0\) as \(\varepsilon \rightarrow 0\). Actually, \(K_{c}\) is a kernel operator, so \(\left \| K_{c}1_{\varepsilon }\right \| =\sup _{l^{\prime }\in \left [ 0,\varepsilon \right ] }\int _{0}^{l_{2}}k(l,l^{\prime })\text{d}l\) and, in general, \(\lim _{\varepsilon \rightarrow 0}\left \| K_{c}1_{\varepsilon }\right \| \) need not be equal to zero.


47D06 One-parameter semigroups and linear evolution equations
92D25 Population dynamics (general)
Full Text: Euclid